A375616 a(n) is the number of lucky cars in all parking functions of order n.

0, 1, 5, 36, 350, 4320, 64827, 1146880, 23383404, 540000000, 13933327265, 397303087104, 12407264266410, 421154777645056, 15439814208984375, 607985949695016960, 25593429637028941208, 1146928904801167933440, 54515427164280400691709, 2739404800000000000000000

Offset: 0

Author

Kimberly P. Hadaway, Aug 21 2024, suggested by Andrew Howroyd

Keywords

• nonn

Comments

This sequence enumerates lucky cars in parking functions of order n (where a lucky spot is one which is parked in by a car which prefers that spot).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386
• Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 10.

Crossrefs

Row sums of A374756.

Cf. A370832.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      expand(x*mul((n+1-k)+k*x, k=2..n)))
    end:
a:= n-> add(k*coeff(b(n), x, k), k=1..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 21 2024

Mathematica

b[n_] := b[n] = If[n == 0, 1, Expand[x*Product[(n+1-k) + k*x, {k, 2, n}]]];
a[n_] := Sum[k*Coefficient[b[n], x, k], {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 31 2024, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k*A370832(n,k) = Sum_{k=1..n} A374756(n,k).

Extensions

a(6)-a(19) from Alois P. Heinz, Aug 21 2024